We investigate constraints on scalar dark matter (DM) by analyzing the Lyman-$\alpha$ forest, which probes structure formation at medium and small scales, and also by studying its cosmological consequences at high and low redshift. For scalar DM that constitutes more than 30% of the total DM density, we obtain a lower limit $m\gtrsim {10}^{-21}\mathrm{eV}$ for the mass of scalar DM. This implies an upper limit on the initial field displacement (or the decay constant for an axion-like field) of $\varphi \lesssim {10}^{16}\mathrm{GeV}$. We also derive limits on the energy scale of cosmic inflation and establish an upper bound on the tensor-to-scalar ratio of $r<{10}^{-3}$ in the presence of scalar DM. Furthermore, we show that there is very little room for ultralight scalar DM to solve the “small-scale crisis” of cold DM without spoiling the Lyman-$\alpha$ forest results. The constraints presented in this paper can be used for testing generic theories that contain light scalar fields.

• Received 8 August 2017

DOI:https://doi.org/10.1103/PhysRevD.96.123514